Spaces without cohomological dimension preserving compactifications

Dydak, Jerzy; Walsh, John J.

doi:10.1090/S0002-9939-1991-1081695-3

Spaces without cohomological dimension preserving compactifications
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by Jerzy Dydak and John J. Walsh PDF

Proc. Amer. Math. Soc. 113 (1991), 1155-1162 Request permission

Abstract:

Examples are constructed that include: first, a separable metric space having cohomological dimension 4 such that every Hausdorff compactification has cohomological dimension at least 5; second, a locally compact metric space having cohomological dimension 4 whose Stone-Čech compactification has infinite cohomological dimension.

References

D. W. Anderson and Luke Hodgkin, The $K$-theory of Eilenberg-MacLane complexes, Topology 7 (1968), 317–329. MR 231369, DOI 10.1016/0040-9383(68)90009-8
V. M. Buhštaber and A. S. Miščenko, A $K$-theory on the category of infinite cell complexes, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 560–604 (Russian). MR 0253322
A. N. Dranishnikov, On a problem of P. S. Aleksandrov, Mat. Sb. (N.S.) 135(177) (1988), no. 4, 551–557, 560 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 2, 539–545. MR 942139, DOI 10.1070/SM1989v063n02ABEH003290
A. N. Dranishnikov, The cohomological dimension is not preserved under the Stone-Čech compactification, C. R. Acad. Bulgare Sci. 41 (1988), no. 12, 9–10 (Russian). MR 985881
A. T. Fomenko, D. B. Fuchs, and V. L. Gutenmacher, Homotopic topology, Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1986. Translated from the Russian by K. Mályusz. MR 873943
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
V. I. Kuz′minov, Homological dimension theory, Uspehi Mat. Nauk 23 (1968), no. 5 (143), 3–49 (Russian). MR 0240813
J. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341. MR 159327

Characterizing cohomological dimension: the cohomological dimension of

James E. West, Open problems in infinite-dimensional topology, Open problems in topology, North-Holland, Amsterdam, 1990, pp. 523–597. MR 1078666